Problem 15 - Algebra 1

Question

Write an equation of the line that passes through the given points. $$ (2,12),(7,2) $$

Step-by-Step Solution

Answer

The equation of the line that passes through the points (2,12) and (7,2) is \(y = -2x + 16\).

Step 1: Calculate Slope

The first step is to find the slope of the line passing through the given points. The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Plugging in the given points (2, 12) and (7, 2), the slope \(m\) becomes \( m = \frac{2 - 12}{7 - 2} = -2\).

Step 2: Point-Slope Form of Equation

Next, use the basic point-slope form equation, which is \(y - y1 = m(x - x1)\), where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope of the line. We can use either of the given points (2, 12) or (7, 2) and the calculated slope -2 to form the equation. Let's use point (2, 12). So, the equation becomes: \( y - 12 = -2(x - 2)\).

Step 3: Y-Intercept form

Finally, convert the point-slope form to slope-intercept form by solving for y: \(y = mx + b\). Simplify the equation \( y - 12 = -2(x - 2)\) to slope-intercept form. This results in \(y = -2x + 16\) which is in slope-intercept form.

Key Concepts

Slope Calculation

Understanding the slope of a line is vital, as it indicates the direction and steepness of the line. In mathematical terms, the slope is a measure of how much the y-coordinate of a point on the line changes with a change in the x-coordinate. To calculate the slope, often denoted as m, you need two points on the line, let's denote them as (x1, y1) and (x2, y2). The formula for finding the slope is:
\[ m = \frac{y2 - y1}{x2 - x1} \]
This formula essentially provides us with the rise over run, or how much we 'rise' (change in y) for a given 'run' (change in x). In the example of the points (2, 12) and (7, 2), subtracting the y-coordinates and then the x-coordinates gives us a slope of:
\[ m = \frac{2 - 12}{7 - 2} = \frac{-10}{5} = -2 \]
This tells us that for every move to the right along the x-axis by one unit, the line drops down by two units, resulting in a downward sloping line since the slope is negative.

Point-Slope Form

Once the slope of a line is known, the line can be uniquely defined using the point-slope form with any point that lies on the line. The point-slope equation is written as:
\[ y - y1 = m(x - x1) \]
where (x1, y1) is the point on the line (you can choose any of the two given points) and m is the slope calculated earlier. Lets take the point (2, 12) as an example, and using our earlier calculated slope m = -2, we can create the equation:
\[ y - 12 = -2(x - 2) \]
This form is especially convenient when you have a point and the slope and you want to quickly write the equation of a line. The point-slope form directly shows the slope and a point, making it very clear to anyone reading the equation.

Slope-Intercept Form

The slope-intercept form is perhaps the most widely used form to represent the equation of a line. It is expressed as:
\[ y = mx + b \]
where m is the slope of the line and b is the y-intercept — the y-coordinate where the line crosses the y-axis. To rewrite the point-slope form (y - 12 = -2(x - 2)) to slope-intercept form, we simplify the equation to look like this:
\[ y = -2x + 16 \]
In the slope-intercept form, the two key pieces of information, the slope and the y-intercept, are immediately clear. This form is highly valuable for easily graphing linear equations or comparing the slopes and interceptions of different lines. Moreover, converting to this form often helps in finding where two lines intersect, or determining if they are parallel.